Inflation Illusion and Stock PricesThe Harvard community has made this

article openly available. Please share howthis access benefits you. Your story matters

Citation Campbell, John Y., and Tuomo Vuolteenaho. 2004. Inflation illusionand stock prices. American Economic Review 94, no. 2: 19-23.

Published Version http://dx.doi.org/10.1257/0002828041301533

Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:3196090

Terms of Use This article was downloaded from Harvard University’s DASHrepository, and is made available under the terms and conditionsapplicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA

NBER WORKING PAPER SERIES

INFLATION ILLUSION AND STOCK PRICES

John Y. CampbellTuomo Vuolteenaho

Working Paper 10263http://www.nber.org/papers/w10263

NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts Avenue

Cambridge, MA 02138January 2004

We would like to thank Cliff Asness for his helpful comments. This material is based upon work supportedby the National Science Foundation under Grant No. 0214061 to Campbell. The views expressed herein arethose of the authors and not necessarily those of the National Bureau of Economic Research.

©2004 by John Y. Campbell and Tuomo Vuolteenaho. All rights reserved. Short sections of text, not toexceed two paragraphs, may be quoted without explicit permission provided that full credit, including ©notice, is given to the source.

Inflation Illusion and Stock PricesJohn Y. Campbell and Tuomo VuolteenahoNBER Working Paper No. 10263January 2004JEL No. G12, G14, N22

ABSTRACT

We empirically decompose the S&P 500's dividend yield into (1) a rational forecast of long-run real

dividend growth, (2) the subjectively expected risk premium, and (3) residual mispricing attributed

to the market's forecast of dividend growth deviating from the rational forecast. Modigliani and

Cohn's (1979) hypothesis and the persistent use of the "Fed model" by Wall Street suggest that the

stock market incorrectly extrapolates past nominal growth rates without taking into account the

impact of time-varying inflation. Consistent with the Modigliani-Cohn hypothesis, we find that the

level of inflation explains almost 80% of the time-series variation in stock-market mispricing.

John Y. CampbellDepartment of EconomicsHarvard UniversityLittauer Center 213Cambridge, MA 02138and NBERjohn_campbell@harvard.edu

Tuomo VuolteenahoHarvard UniversityDepartment of EconomicsLittauer CenterCambridge, MA 02138and NBERt_vuolteenaho@harvard.edu

The “Fed model” of equity valuation

The influence of the macroeconomy on the stock market is a subject of enduringimportance and fascination to academics, investment professionals, and monetarypolicymakers. Academics have devoted much of their effort to models that link stockprices to consumption through the first-order conditions of individual investors. Mostinvestment professionals have adopted a radically different perspective. The leadingpractitioner model of equity valuation, the so-called “Fed model,” relates the yield onstocks (as measured by the ratio of dividends or earnings to stock prices) to the yieldon nominal Treasury bonds. The idea is that stocks and bonds compete for spacein investors’ portfolios. If the yield on bonds rises, then the yield on stocks mustalso rise to maintain the competitiveness of stocks. The model is often augmented toinclude a measure of the relative risk premium on stocks versus bonds, for example,the historical relative volatility of the returns on these two asset classes. Practitionersargue that the bond yield plus a risk premium defines a “normal” yield on stocks, andthat the actual stock yield tends to revert to this normal yield. If the measured stockyield exceeds the normal yield defined by the Fed model, then stocks are attractivelypriced; if the measured yield falls below the normal yield, then stocks are overpriced.

Historically, the major influence on nominal bond yields has been the rate ofinflation. Thus the Fed model implies that stock yields are highly correlated withinflation. In the late 1990’s practitioners often argued that falling stock yields, andrising stock prices, were justified by declining inflation.

As pointed out by Asness (2000, 2003), the Fed model has been quite successfulas an empirical description of stock prices. Most notably, the model describes therise in stock yields, along with inflation, during the 1970’s and early 1980’s, and thedecline in stock yields during the past 20 years. Of course, movements in stockand bond prices are highly persistent, and so it is desirable to evaluate the empiricalperformance of the Fed model over a longer period. In Table 1 we use a series ofregressions to illustrate the performance of the model in the period 1927-2002. Eachregression has a measure of the stock yield as the dependent variable, and proxiesfor the risk premium and the nominal discount rate as the independent variables.Our basic measure of the stock yield is the dividend-price ratio, but we also considerthe earnings-price ratio, smoothed by averaging earnings over five years. Our firstproxy for the risk premium is a cross-sectional beta premium within the stock market,calculated by Polk, Thompson, and Vuolteenaho (2003) as the rank-order correlationof stock-level yield measures with the stock’s CAPM beta. When this correlation

1

is high, it implies that market participants expect a high reward for bearing marketrisk since riskier (higher-beta) stocks are much cheaper than less risky stocks. Oursecond risk premium proxy is the historical volatility of excess stock returns relativeto that of nominal bonds, a measure previously employed by, e.g., Asness (2003).To measure the nominal discount rate we use either the average historical rate ofinflation, calculated as an exponentially declining moving average of past monthlyinflation rates, or the nominal yield on ten-year Treasury bonds.

The benchmark regression in the first row of Table 1 explains 49% of the variationin the dividend-price ratio using the cross-sectional risk premium and the histori-cal inflation rate. Both variables are strongly statistically significant. Beyond thestatistical significance, the fit of this regression is illustrated in Figure 1. Here thetriangles are the demeaned dividend-price ratio, the solid line is the contribution ofthe cross-sectional risk premium to the fit of the regression (the multivariate regres-sion coefficient times the risk premium), and the dashed line is the contribution of theinflation rate to the fit of the regression. The figure shows that much of the volatilemovement of the dividend-price ratio during the 1930’s and 1940’s was related tovolatile movement of the cross-sectional risk premium. At the bottom of the GreatDepression, this explained the high level of the dividend-price ratio despite the pres-ence of deflation, which our estimated model suggests should have supported stockprices. As inflation rose during the late 1930’s and 1940’s, its negative influence onstock prices was outweighted by the declining risk premium. In the postwar periodthe cross-sectional risk premium generally trended downward, reaching a low point inthe early 1980’s. The inflation rate rose steadily during the 1960’s and 1970’s, andaccounts for the high dividend-price ratio (depressed stock prices) of the early 1980’s.During the later 1980’s and 1990’s, the dividend-price ratio was driven downwardsboth by a declining cross-sectional equity premium and by a declining inflation rate,although the run-up in stock prices during the 1990’s was too extreme to be fullyexplained by our two variables.

Alternative specifications are reported in subsequent rows of Table 1. Row twouses the five-year smoothed earnings yield rather than the dividend yield; this changereduces the R2 substantially, but has little effect on the estimated coefficients. Aregression that substitutes relative volatility for the cross-sectional risk premium (rowthree) has a very similar fit to the benchmark regression. Substituting the nominalbond yield for historical inflation reduces the fit somewhat if the risk measure isthe cross-sectional risk premium (row four), but not if the risk measure is relativevolatility (row five).

2

Interpreting the Fed model

Despite the empirical success of the Fed model as a behavioral description of stockprices, there is a serious difficulty with this model as a rational explanation of stockprices (Modigliani and Cohn 1979, Ritter and Warr 2002, Asness 2000, 2003). Tounderstand the difficulty, consider the classic “Gordon growth model” (Williams 1938,Gordon 1962) that expresses the dividend-price ratio in steady state as

DtPt−1

= R−G, (1)

where R is the long-term discount rate and G is the long-term growth rate of divi-dends. The Fed model argues that the discount rate on stocks is the yield on bondsplus a proxy for the risk premium of stocks over bonds. These two variables willexplain the movement in stock prices if G is constant.

The problem with this interpretation is that the main influence on the long-termnominal interest rate is the expected long-term rate of inflation (Fama 1975, 1990,Mishkin 1990a, 1990b). The long-term real interest rate is comparatively stableand does not move closely with the long-term nominal rate. While the Gordongrowth model can be written in nominal terms, this requires that dividend growthG is measured in nominal terms. Since stocks are claims to the productive capitalof the real economy, one would expect that a change in long-term expected inflationwould move nominal G one-for-one, offsetting the effect on nominal R and leavingthe dividend-price ratio unaffected. In real terms, neither R nor G should changewith expected inflation. This criticism of the Fed model is forcefully expressed byAsness (2003).

If the above conventional argument does not explain the empirical relation betweenstock prices and inflation, what does explain this relation? One possibility is thatinflation (or the monetary authority’s response to inflation) damages the real economy,and particularly the profitability of the corporate sector. In this case real G mightfall when inflation rises, justifiably driving up the dividend-price ratio. A variant ofthis idea reverses the causality, arguing that poor economic prospects induce fiscaland monetary responses that ultimately increase inflation (Geske and Roll 1983).The difficulty with this argument is that G is correctly interpreted as a long-termreal dividend growth, not the conditional expected growth at business-cycle horizonssuch as three years. A second possibility is that inflation makes investors more riskaverse, driving up the equity premium and thus the real discount rate R. Brandtand Wang (2003) present a model of this effect.

3

Modigliani and Cohn (1979) propose a more radical third hypothesis. They claimthat stock market investors (but not bond market investors) are subject to inflationillusion. Stock market investors fail to understand the effect of inflation on nom-inal dividend growth rates and extrapolate historical nominal growth rates even inperiods of changing inflation. Thus when inflation rises, bond market participantsincrease nominal interest rates which are used by stock market participants to dis-count unchanged expectations of future nominal dividends. The dividend-price ratiomoves with the nominal bond yield because stock market investors irrationally failto adjust the nominal growth rate G to match the nominal discount rate R. Fromthe perspective of a rational investor, this implies that stock prices are undervaluedwhen inflation is high, and may become overvalued when inflation falls. The div-idend yield that emerges from the interaction of rational and irrational investors ispositively correlated with inflation and the long-term nominal interest rate. In recentwork, Ritter and Warr (2002) support this idea with a detailed empirical analysis ofthe 1983-2000 bull market.

In this paper we try to distinguish among these three alternative views. Tounderstand our approach, it is helpful to return to the simple Gordon growth modeland subtract the riskless interest rate from both the discount rate and the growthrate of dividends. We define the excess discount rate as Re ≡ R−Rf and the excessdividend growth rate as Ge ≡ G−Rf . We are considering the possibility that someinvestors are irrational, so we must distinguish between the subjective expectationsof irrational investors and the objective expectations of rational investors. As longas the irrational investors simply use the present value formula with an erroneousexpected growth rate, both sets of expectations must obey the accounting identity ofthe Gordon growth model. Thus we have

D

P= Re,OBJ −Ge,OBJ = Re,SUBJ −Ge,SUBJ (2)

= −Ge,OBJ +Re,SUBJ + (Ge,OBJ −Ge,SUBJ).In words, the dividend yield has three components: (1) the negative of objectively ex-pected excess dividend growth, (2) the subjective risk premium, and (3) a mispricingterm that is due to a divergence between the objective (i.e, rational) and subjective(i.e, irrational) growth forecast.

The first step in our analysis is to show that Ge,OBJ = Re,OBJ − D/P tends torise, not fall, and that Re,SUBJ tends to fall, not rise, with inflation, thus ruling outthe rational justifications for the comovement of the dividend yield with inflation.

4

The second step is to show that, consistent with the Modigliani-Cohn view, highinflation coincides with a positive divergence between objective and subjective growthexpectations Ge, the wedge between the growth expectations of rational investors andirrational investors who are subject to inflation illusion.

Empirical implementation and results

In our empirical implementation we use the loglinear dynamic valuation frame-work of Campbell and Shiller (1988), since this framework, unlike the simple Gordonmodel, allows for time-varying discount rates and dividend growth rates. The Gor-don model (1) has the limitation that the expected returns and expected growthmust be constant, thus it is suitable for our purposes only as an illustration. TheCampbell-Shiller formula for the log dividend-price ratio can be written as

dt−1 − pt−1 ≈ k

ρ− 1 +∞Xj=0

ρjEt−1ret+j −∞Xj=0

ρjEt−1∆det+j, (3)

where ∆d denotes log dividend growth, r denotes log stock return, ∆de denotes ∆dless the log risk-free rate for the period, and re denotes r less the log risk-free rate forthe period. ρ and k are parameters of linearization defined by ρ ≡ 1±¡1+exp(d− p)¢and k ≡ − log(ρ) − (1 − ρ) log(1/ρ − 1). A typical value for ρ is 0.97. When thedividend-price ratio is constant, then ρ = P/(P +D), the ratio of the ex-dividend tothe cum-dividend stock price. Comparing Equations (2) and (3),

P∞j=0 ρ

jEt−1ret+j isanalogous to Re,OBJ and Re,SUBJ , and

P∞j=0 ρ

jEt−1∆det+j is analogous to Ge,OBJ and

Ge,SUBJ , depending on whether the expectations taken in Equation (3) are objective(i.e., rational) or subjective (i.e., irrational).

In our empirical work, we will estimate the termP∞

j=0 ρjEOBJt−1 r

et+j under objective

expectations. The objective expected growth can then be inferred from Equation (3)as k

ρ−1 +P∞

j=0 ρjEOBJt−1 r

et+j − (dt−1− pt−1). The subjective risk premium is estimated

as the fitted value of a regression ofP∞

j=0 ρjEOBJt−1 r

et+j on a subjective risk-premium

proxy λt (for example, on the relative volatility of stocks vs. that of bonds):

∞Xj=0

ρjEOBJt−1 ret+j = constant+ γλt + εt (4)

Mispricing, or the difference between objective and subjective expected dividendgrowth, is the residual εt of this regression. When stocks are subjectively perceived

5

to be very risky, then the fitted value γλt is high. In contrast, when stocks are un-derpriced the residual εt is high. Selecting γ with a regression implies an assumptionthat mispricing is independent of the subjective risk premium. This choice for γ isin a sense conservative since it minimizes the variance of mispricing.

Following Campbell (1991) and Campbell and Ammer (1993), we combine thevaluation framework with a vector autoregression (VAR) that predicts stock returns.The VAR includes the excess log return on S&P 500 index over the three-month Trea-sury bill (re), the cross-sectional beta premium of Polk, Thompson, and Vuolteenaho(2003, λSRC), the log dividend-price ratio (dy), and the exponentially smoothed mov-ing average of inflation (π). The sample period for the dependent variables is 1927:6-2002:12, and the sample thus consists of 303 quarterly data points. Table 2 showsthe estimated parameters of the VAR system. We use this VAR to infer objectiveand subjective excess dividend growth and returns.

Table 3 shows the standard deviations and Figure 2 a time-series plot of the threeestimated components of the log dividend yield: −P∞

j=0 ρjEOBJt−1 ∆det+j, γλ

SRCt , and

εt. Together the three series add up to log-dividend yield. According to Table 3,mispricing is the most volatile component of the log dividend-price ratio, while thesubjective risk component is the least volatile. Table 3 also shows the functions thatmap the four VAR state variables into the three components of the log dividend priceratio. Dividends are expected to grow rapidly when the subjective risk premiumand/or inflation is high. Objectively expected excess returns on stocks are highwhen the subjective risk premium, dividend yield, and/or inflation are high.

Below each number in Table 3 is a standard error, calculated from 10,000 bootstrapsimulations of the VAR in Table 2, and an estimate of bias, calculated from the sameset of simulations. These standard errors and bias estimates must of course be treatedwith caution since they are conditional on the validity of the estimated VAR as adescription of the data generating process. However, they provide some evidencethat the volatility of mispricing, and the effects of inflation on expected dividendgrowth and mispricing, are statistically significant and are if anything understated byour point estimates.

Perhaps surprisingly, our model detects stock prices significantly overpriced inthe summer of 1933, although the dividend-price ratio is approximately at its sampleaverage. This detected overpricing can be explained by two simultaneous phenomena:First, the expected long-run dividend growth was extremely low at 2.5 standarddeviations below its sample mean. Second, the subjective risk premium was 1.3

6

standard deviations above its unconditional mean. Together, these two facts wouldhave predicted a high dividend yield and low stock prices, contrary to the aboutaverage dividend yield observed at the time. Perhaps not surprisingly, another timewhen stock prices are detected as overpriced is the end of 1999, a point in time manyconsider the top of the technology bubble. Although at the end of 1999 the subjectiverisk premium was 0.74 standard deviations below its unconditional mean, expecteddividend growth was close to its unconditional mean. The two variables togetherjustify a dividend-price ratio of 3.3% versus the observed 1.2%, leaving a gap of 2.1%to be explained by mispricing. At the other end of the spectrum, our VAR detectsstock as overpriced in 1947 and 1983, years in which dividend yields weree more thantwo and half times our fair yield estimates.

The central prediction of the Modigliani-Cohn (1979) hypothesis is that high in-flation leads to stock market underpricing and low (or negative) inflation leads tooverpricing. The main competing hypotheses are that low stock prices coincidingwith high inflation are rationally justified because high inflation coincides with lowexpected dividend growth or a high subjective risk premium. To examine these hy-potheses, we regress the three components of dividend yield on smoothed inflation inTable 3. The regression coefficient of −P∞

j=0 ρjEOBJt−1 ∆det+j on inflation is -11.25 with

an R2 of 95%, implying a positive, not negative, relation between rationally expectedexcess dividend growth and inflation. To the extent that inflation coincides withpoor economic conditions, the worst seems to be over and future growth looks bright(perhaps because of the low initial level). The subjective risk premium seems largelyunrelated to inflation. Thus, we reject the rational hypotheses justifying the positiveassociation of dividend yield and inflation.

In contrast, our VAR results in Table 3 provide strong support to the Modigliani-Cohn (1979) hypothesis. The regression coefficient of εt on inflation is stronglypositive, and statistically and economically significant. The strength of this relationis illustrated in Figure 3 that plots mispricing, εt, and inflation scaled by its regressioncoefficient, against time. The 78% R2 is evident in the figure as the lines plot almostperfectly on top of each other, as predicted by Modigliani and Cohn.

Table 4 examines the robustness of these results to minor variations in the VARspecification. The first row of the table show the main results from Table 3 asa basis for comparison. Rows two and three alter the data frequency to monthlyand annual, with little changes in our main results. Row four uses the relativevolatility of stocks vs. bonds as the subjective risk measure, significantly reducing

7

the positive correlation between expected excess dividend growth and inflation butactually increasing the R2 of inflation on mispricing. Row five substitutes five-yearsmoothed earnings yield for dividend yield, row six adds the TERM yield spread (theyield difference between long-term and short-term government bonds) to the VAR,and row seven defines excess returns and excess dividend growth rates relative tolong-term bond holding-period returns, again with little impact on our main results.Adding three additional lags to our VAR specification adds noise to the estimates; italmost eliminates the positive correlation between dividend growth and inflation, butreduces the R2 of inflation on mispricing only to 49%. Finally, row nine accountsfor the possibility that the conditional variance of returns may systematically movewith the VAR state variables and thus create a time-varying wedge between expectedsimple and log returns, with no change in results.

Conclusion

This paper presents a three-way decomposition of the stock market dividend yieldinto a term due to rationally expected long-run dividend growth, a term due to thesubjective risk premium on the market, and a residual term that we attribute to adeviation of subjectively expected dividend growth from objectively expected growth.We use a VAR system to construct empirical estimates of these three components.We find that high inflation is positively correlated with rationally expected long-runreal dividend growth; thus the negative effect of inflation on stock prices cannot beexplained through this channel. We find that inflation is almost uncorrelated withthe subjective risk premium. However, inflation is highly correlated with mispricing,supporting the Modigliani-Cohn (1979) view that investors form subjective growthforecasts by extrapolating past nominal growth rates without adjusting for changesin inflation.

In interpreting this result, it is important to keep in mind that inflation, thestock market dividend yield, and its estimated components are all highly persistentprocesses. Thus we have relatively few independent observations even in a 75-yearhistory, and our results should be regarded as suggestive rather than decisive. Infuture work it will be important to gather additional evidence by examining therelation between inflation and stock returns in other countries. It may also behelpful to use survey data to measure subjective inflation expectations. In addition,we cannot rule out the possibility that some part of what we call mispricing is in facta second component of the subjective risk premium, one that is common to all stocksand thus does not appear in our cross-sectional measure of risk.

8

The Modigliani-Cohn hypothesis has interesting implications both for investorsand for monetary policymakers. Investors need to know whether stocks, as realassets, provide a hedge against inflation. Fama and Schwert (1977) and othershave documented the negative effect of inflation shocks on realized stock returns.The Modigliani-Cohn hypothesis explains this as the result of mispricing driven byinflation illusion, an effect which should diminish over the longer run. Boudoukh andRichardson (1993) examine this issue directly and find that stocks are better inflationhedges over five-year periods than over one-year periods.

There has been an active recent debate about whether monetary policy should beused to combat stock market mispricing. The Modigliani-Cohn hypothesis suggeststhat disinflation may itself generate mispricing by confusing stock market investorswho are subject to inflation illusion. It also implies that a successful stabilizationof inflation will reduce the volatility of mispricing and thereby contribute to theefficiency of the stock market.

9

References

Asness, Clifford, 2000, Stocks versus bonds: explaining the equity risk premium,Financial Analysts Journal, March/April 2000, 96-113.

Asness, Clifford, 2003, Fight the Fed model: the relationship between future returnsand stock and bond market yields, Journal of Portfolio Management, Fall 2003,11-24.

Boudoukh, Jacob and Matthew Richardson, 1993, Stock returns and inflation: Along-horizon perspective, American Economic Review 83, 1346—1355.

Brandt, Michael W. and Kevin Q. Wang, 2003, Time-varying risk aversion andunexpected inflation, Journal of Monetary Economics 50, 1457—1498.

Campbell, John Y., 1991, A variance decomposition for stock returns, EconomicJournal 101, 157—179.

Campbell, John Y. and John Ammer, 1993, What moves the stock and bond mar-kets? A variance decomposition for long-term asset returns, Journal of Finance48, 3—37.

Campbell, John Y. and Robert J. Shiller, 1988, The dividend-price ratio and expec-tations of future dividends and discount factors, Review of Financial Studies 1,195—228.

Fama, Eugene F., 1975, Short-term interest rates as predictors of inflation, AmericanEconomic Review 65, 269—282.

Fama, Eugene F., 1990, Term-structure forecasts of interest rates, inflation, and realreturns, Journal of Monetary Economics 25, 59—76.

Fama, Eugene F. and G. William Schwert, 1977, Asset returns and inflation, Journalof Financial Economics 5, 115—146.

Geske, Robert and Richard Roll, 1983, The fiscal and monetary linkage betweenstock returns and inflation, Journal of Finance 38, 1—33.

Gordon, Myron, 1962, The Investment, Financing, and Valuation of the Corporation,Irwin, Homewood, IL.

10

Mishkin, Frederic, 1990a, The information in the longer-maturity term structureabout future inflation, Quarterly Journal of Economics 105, 815—821.

Mishkin, Frederic, 1990b, What does the term structure tell us about future infla-tion?, Journal of Monetary Economics 25, 77—95.

Modigliani, Franco and Richard Cohn, 1979, Inflation, rational valuation, and themarket, Financial Analysts’ Journal.

Polk, Christopher, Samuel Thompson, and Tuomo Vuolteenaho, 2003, New fore-casts of the equity premium, unpublished paper, Northwestern University andHarvard University.

Ritter, Jay R. and Richard S.Warr, 2002, The decline of inflation and the bull marketof 1982—1999, Journal of Financial and Quantitative Analysis 37, 29—61.

John Burr Williams, 1938, Evaluation by the rule of present worth, The Theory ofInvestment Value, 55-75.

11

Table 1: Explaining the stock yield with a subjective risk-premium measure andnominal bond yieldThe table shows the OLS regressions of stock yield (either dividend yield, DY , orsmoothed earnings yield where earnings are averaged over the past five years, 5EY )on a subjective risk premium measure (either the Spearman rank correlation betweenfirms’ valuation multiples and estimated CAPM beta, λSRC , or relative past volatilityof stocks relative to bonds, λRV ) and on a nominal yield measure (either the ten-yeartaxable Treasury bond yield, LTY , or smoothed past inflation, π). The t-statisticsare computed using Newey-West (1987) standard errors with 60 leads and lags. R2

is adjusted for the degrees of freedom. The regressions are estimated from the fullsample period 1927:5-2002:12, 908 monthly observations.

Stock Risk Nominal adj.yield Const. measure bond yield R2

DYt = 0.0397 +0.0512 × λSRCt +0.2146 × πt 49.2%(15.2) (6.4) (3.1)

5EYt = 0.0647 +0.0684 × λSRCt +0.2381 × πt 27.5%(8.2) (3.4) (1.2)

DYt = 0.0165 +0.0029 × λRVt +0.2156 × πt 49.2%(3.4) (5.9) (2.6)

DYt = 0.0378 +0.0547 × λSRCt +0.1528 × LTYt 37.2%(4.0) (5.2) (1.5)

DYt = -0.0069 +0.0046 × λRVt +0.3576 × LTYt 49.1%(-1.0) (6.6) (5.6)

12

Table 2: VAR parameter estimatesThe table shows the OLS parameter estimates for a first-order VAR model includinga constant, the log excess market return (reM), the subjective risk-premium measure(λSRC), log dividend-price ratio (dy), and smoothed inflation (π). Each set of threerows corresponds to a different dependent variable. The first five columns report co-efficients on the five explanatory variables, and the last column shows R2. Bootstrapstandard errors (in parentheses) are computed from 10,000 realizations simulated fromthe estimated system. The table also reports the correlation matrix of the shockswith shock standard deviations on the diagonal, labeled “corr/std.” Sample periodfor the dependent variables is 1927:6-2002:12, 303 quarterly data points.

constant reM,t λSRCt dyt πt R2 %

reM,t+1 0.0768 -0.0520 0.0377 0.0190 0.1556 1.94(0.0902) (0.0592) (0.0480) (0.0266) (0.4208)

λSRCt+1 0.2501 0.0069 0.8160 0.07653 -0.6469 86.70(0.0612) (0.0434) (0.0342) (0.0180) (0.2796)

dyt+1 -0.1402 0.0587 0.0087 0.9609 0.3370 92.67(0.0943) (0.0610) (0.0496) (0.0278) (0.4428)

πt+1 -0.0036 0.0007 0.0026 -0.0012 0.9966 97.59(0.0039) (0.0026) (0.0020) (0.0011) (0.0196)

corr/std reM,t+1 λSRCt+1 dyt+1 πt+1

reM,t+1 0.1084(0.0091)

λSRCt+1 0.0227 0.0779(0.1143) (0.0069)

dyt+1 -0.9063 -0.0132 0.1109(0.0239) (0.1082) (0.0082)

πt+1 0.1290 -0.1658 -0.0983 0.0047(0.0986) (0.0836) (0.0991) (0.0003)

13

Table 3: Regressions of dividend yield’s components on inflationThe table shows derived statistics implied by the VAR model of Table 2. We decom-pose the demeaned log dividend yield, dyt, into three components: (1) The negativeof long-run expected dividend growth, −P∞

i=1 EOBJt ∆det+1, where ∆det+1 is the de-

meaned dividend growth; (2) the subjective risk-premium component, γλSRCt ; and(3) the mispricing component. The subjective risk-premium and mispricing compo-nents are defined as the fitted values and residuals of the regression

P∞i=1 E

OBJt ret+1 =

γλSRCt + εt, where ret+1 is the demeaned excess log return on S&P 500 and λSRCt thedemeaned cross-sectional beta premium. The upper-left section of the table, labeled“Stdev” shows the unconditional standard deviations of the three components. Thelower left section shows the linear functions that map the VAR state variables into thethree components of demeaned log dividend yield. The VAR state variables are thelog excess S&P 500 return (reM), the cross-sectional beta premium measure (λSRC),log dividend-price ratio (dy), and smoothed inflation (π). The right half of the tableshows the simple regression coefficients of log dividend yield and its three componentson smoothed inflation (π) and the corresponding regression R2s. Standard errors (inparentheses) and bias estimates [in brackets] are computed from 10,000 simulationsfrom the VAR of Table 2.

dyt = −PE∆de +γλSRCt +εt Regressions on inflationStdev: 0.3446 0.1831 0.5689(s.e) (0.3212) (0.2665) (0.2966)[bias] [-0.0910] [0.0200] [-0.18880] Dependent Coefficient

Function: −PE∆de +γλSRCt +εt variable: on πt: R2%

reM,t -0.0040 0 0.0040 dyt 4.01 7.19(0.0455) (N/A) (0.0455) (6.02) (14.48)[0.0050] [N/A] [-0.0050] [-2.34] [4.74]

λSRCt -0.4120 0.8342 -0.4222 −PE∆de -11.25 94.78(0.1799) (1.0399) (1.0367) (6.35) (33.46)[0.0955] [-0.0431] [-0.0524] [4.24] [-36.01]

dyt 0.2506 0 0.7493 +γλSRCt -1.58 6.61(0.2320) (N/A) (0.2320) (3.85) (15.42)[-0.0085] [N/A] [0.0085] [0.25] [9.65]

πt -13.0318 0 13.0318 +εt 16.83 77.90(6.2212) (N/A) (6.2212) (6.78) (27.07)[5.2482] [N/A] [-5.2482] [-6.84] [-35.33]

14

Table 4: Alternative VAR specificationsThe table examines the robustness of the implied regression statistics presented inTable 3. In the base specification, we first decompose the demeaned log dividendyield, dyt, into three components: (1) The negative of long-run expected dividendgrowth, −P∞

i=1 EOBJt ∆det+1, where ∆d

et+1 is the demeaned excess dividend growth;

(2) the subjective risk-premium component, γλt; and (3) the mispricing component.The subjective risk-premium and mispricing components are defined as the fittedvalues and residuals of the regression

P∞i=1 E

OBJt ret+1 = γλSRCt +εt, where ret+1 is the

demeaned excess log return on S&P 500 and λSRCt the demeaned cross-sectional betapremium. Then we compute the simple regression coefficients of log dividend yieldand its three components on smoothed inflation (π) and the corresponding regressionR2s. Each row in the table corresponds to small change in the VAR specification ofTables 2 and 3.

VAR specification: −PE∆de R2% +γλt R2% +εt R2%

1. Base case -11.25 94.78 -1.58 6.61 16.83 77.902. Monthly data -11.93 92.91 -1.40 7.15 17.10 80.073. Annual data -6.47 67.75 -3.16 10.45 12.87 64.264. λRV instead of λSRC -0.60 0.11 -4.17 2.75 10.15 82.085. 5ey instead of dy -8.18 95.19 -1.42 9.28 12.44 61.806. TERM added -10.37 90.69 -1.49 8.13 15.59 75.787. Stocks - bonds -14.65 96.49 -1.36 6.44 20.11 82.508. Four VAR lags -0.55 1.15 -3.13 10.12 7.38 48.639. Jensens’s inequality -11.31 95.22 -1.07 9.35 15.26 72.52

15

1927 1937 1947 1957 1967 1977 1987 1997

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

Dem

eane

d yi

eld

Year

Figure 1: Dividend yield, subjective risk premium, and inflation.

This figure plots the time-series of three variables: (1) Demeaned dividendyield on S&P 500 index, marked with triangles; (2) demeaned subjective risk-premium measure computed by Polk, Thompson, and Vuolteenaho (2003) as thecross-sectional Spearman rank correlation between firms’ valuation levels andestimated betas, marked with a solid line; and (3) the demeaned smoothed inflationcomputed from the producer price index, marked with a dashed line. Subjectiverisk premium and inflation are multiplied by their respective multiple regressioncoefficients in the regression of dividend yield on the subjective risk premium andinflation. The sample period is 1927:5-2002:12.

16

1927 1937 1947 1957 1967 1977 1987 1997-1.5

-1

-0.5

0

0.5

1

1.5

Com

pone

nts

of lo

g(D

/P)

Year

Figure 2: The three components of log dividend yield.

This figure plots the time-series of three variables that add up to demeanedlog dividend yield: (1) deviation of the long-run expected excess dividend growthfrom it’s unconditional mean, marked with a dotted line; (2) demeaned subjectiverisk-premium, marked with a solid line; and (3) the mispricing component, markedwith circles and a line. The subjective risk premium component and the mispricingcomponents are produced by regressing the discounted sum of future expected returnson the subjective risk-premium proxy. The subjective risk-premium component isthe fitted value of this regression, and the mispricing component is the error term ofthis regression. The series are computed using the VAR of Table 2.

17

1927 1937 1947 1957 1967 1977 1987 1997-2

-1.5

-1

-0.5

0

0.5

1

1.5

Mis

pric

ing

and

infla

tion

Year

Figure 3: Inflation and mispricing.

This figure plots the time-series of two variables: (1) The mispricing compo-nent of log dividend yield, marked with circles; and (2) the fitted value from aregression of mispricing component on inflation, marked with a line. The mispricingcomponents is produced as the residual of the regression of discounted sum of futureexpected returns on the subjective risk-premium proxy. The series are computedusing the VAR of Table 2.

18